{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-super-localized-generalized-finite-element","title":"A Super-Localized Generalized Finite Element Method","arxiv_id":"2211.09461","date":"2022-11-17","proceeding":null,"authors":["Philip Freese","Moritz Hauck","Tim Keil","Daniel Peterseim"],"abstract":"This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method's basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method's applicability for challenging high-contrast channeled coefficients.","url_abs":"https://arxiv.org/abs/2211.09461v1","url_pdf":"https://arxiv.org/pdf/2211.09461v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-super-localized-generalized-finite-element","repo_url":"https://github.com/tikeil/sl-gfem","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}